Stochastic analyses arising from a new approach for closed queueing networks

摘要

Analyses are addressed for a number of problems in queueing systems andstochastic modeling that arose due to an investigation into techniques that couldbe used to approximate general closed networks.In Chapter II, a method is presented to calculate the system size distribution atan arbitrary point in time and at departures for a (n)/G/1/N queue. The analysisis carried out using an embedded Markov chain approach. An algorithm is alsodeveloped that combines our analysis with the recursive method of Gupta and Rao.This algorithm compares favorably with that of Gupta and Rao and will solve somesituations when Gupta and Rao's method fails or becomes intractable.In Chapter III, an approach is developed for generating exact solutions of thetime-dependent conditional joint probability distributions for a phase-type renewalprocess. Closed-form expressions are derived when a class of Coxian distributionsare used for the inter-renewal distribution. The class of Coxian distributions waschosen so that solutions could be obtained for any mean and variance desired in theinter-renewal times.In Chapter IV, an algorithm is developed to generate numerical solutions forthe steady-state system size probabilities and waiting time distribution functions ofthe SM/PH/1/N queue by using the matrix-analytic method. Closed form results are also obtained for particular situations of the preceding queue. In addition, itis demonstrated that the SM/PH/1/N model can be implemented to the analysisof a sequential two-queue system. This is an extension to the work by Neuts andChakravarthy.In Chapter V, principal results developed in the preceding chapters are employedfor approximate analysis of the closed network of queues with arbitrary servicetimes. Specifically, the (n)/G/1/N queue is applied to closed networks of ageneral topology, and a sequential two-queue model consisting of the (n)/G/1/Nand SM/PH/1/N queues is proposed for tandem queueing networks.